Newton’s formula for 𝔤𝔩_{𝔫}

Abstract

Abstract. This paper presents an explicit relation between the two sets which are well-known generators of the center of the universal enveloping algebra U (gl n ) of the Lie algebra gl n : one by Capelli (1890) and the other by Gelfand (1950). Our formula is motivated to give an exact analogy for the classical Newton's formula connecting the elementary symmetric functions and the power sum symmetric functions. The formula itself can be deduced from a more general result on Yangians obtained by Nazarov. Our proof is elementary and has an advantage in its direct accessibility.

Mixed tensor products and Capelli-type determinants

“…Note that T is a formal variable that commutes with all E ij , and C n (T ) and C n+1 (T ) will be treated as polynomials in this formal variable. We also note that the polynomials C n+1 (T ) appear in [9] and [11] with a slight change. Namely, the polynomial…”

“…, N also form a system of generators of Z(gl(N)), sometimes known as the Gelfand generators. There is a nice transition formula between the Gelfand generators and the Capelli generators, [11], and this formula can be considered as a noncommutative version of the Newton identities. The applications of Capelli-type determinants extend well beyond relations and properties of elements in Z(gl(N)).…”

On certain conformally invariant systems of differential equations II: Further study of type A systems

“…Lemma 5.2 and Lemma 5.3 are due to Umeda (Lemma 1 and Proposition 2 in [19]). However, due to the interchange of indices and a difference of normalization, our versions look slightly different, so it seems safest to sketch the proof of the more substantial of the two.…”

“…We use three different techniques to study σ(F ), which is why Section 5 involves much more case-by-case argument than the other sections. The first technique is based upon Umeda's Capelli adjoint identity [19]. This is a noncommutative analogue of the method of finding the eigenvalues of an operator by considering its characteristic polynomial and so is very natural.…”

Спектральная Параметризация Для Степенных Сумм Квантовых Суперматриц

“…14)где использовано сокращенное обозначение для разбиений[m | n] k := (n + 1) k , n m−k , [m | n] r := (n m , r). Первое из этих соотношений в предельном случае M(R, R) q→1−→ U (gl n ) было также выведено в работе[11].…”